 School of Mathematics and Statistics  Theses
School of Mathematics and Statistics  Theses
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ItemFast inference for highdimensional onefactor copula models and methods for multivariate changepoint detection with applications to stock market dataVerhoijsen, Alex ( 202311)Copula models are often used to model dependence between random variables in a multivariate, lowdimensional, setting. However, practical problems arise when the dimension becomes very large. A different problem, when working with multivariate time series, is the assumption of stationarity, which is often violated, and adequate methods are required to detect departures from stationary. We begin the thesis by introducing the technical tools related to copula models and changepoint detection that are used in the remainder of this work. The first original contribution of the thesis is a fast inference methods for high dimensional onefactor copula models. We propose a model with Gaussian factors and residual dependence modelled using a onefactor copula model. The corresponding estimation procedure allows the statistician to identify the factor model parameters, the Gaussian factors, the factor in the onefactor copula model, and the onefactor copula parameter. Asymptotic properties for a model with up to three Gaussian factors are established, and finite sample properties are illustrated using Monte Carlo simulations. A practical application using SP500 stock market data demonstrates how the proposed model can be used to capture dependence in realworld data. The second original contribution is a nonparametric sequential openend changepoint detection scheme using the empirical distribution function of possibly multivariate observations. We establish the asymptotic properties of the detector, and perform largescale Monte Carlo simulations in a univariate and multivariate, lowdimensional, setting. To illustrate the procedure, we conclude with a realworld application by monitoring for changes in the logreturns of the NASDAQ composite index. The code used to implement the monitoring procedure is included in the R package npcp.

ItemCombinatorics of Symmetric Functions through Lattice ModelsGunna, Ajeeth Reddy ( 202401)Symmetric functions emerge in many fields of mathematics, serving as characters in representation theory, polynomial representatives in the cohomology rings of varieties in algebraic geometry, and generating series in enumerative combinatorics, among others. Beyond these connections, the ring of symmetric functions itself contains a rich combinatorial theory that propels their systematic study. This PhD thesis explores the application of exactly solvable lattice models to symmetric functions in four distinct papers. Paper 1: Vertex models of canonical Grothendieck polynomials and their duals We study exactly solvable lattice models associated to canonical Grothendieck polynomials and their duals. We derive inversion relations and Cauchy identities. Paper 2: Crystals and integrable systems for edge labeled tableaux We define an integrable five vertex model whose partition function is the generating function $E^{\lambda}$ of edge labeled tableau of shape $\lambda$. Using this, we prove a Cauchytype identity. We give a crystal structure on edge labeled tableau to give a positive Schur expansion of $E^{\lambda}$. Paper 3: Littlewood–Richardson Coefficients of spin Hall–Littlewood Functions We provide a combinatorial formula for the Littlewood–Richardson (LR) coefficients of spin Hall–Littlewood functions, and factorial versions of them. This is achieved by representing these functions and the LR coefficients as the partition function of a lattice model and applying the underlying YangBaxter equation. Our combinatorial expression is in terms of generalised honeycombs; the latter were introduced by Knutson and Tao for ordinary LR coefficients and applied to the computation of Hall polynomials by Zinn–Justin. Paper 4: Shuffle algebras, Lattice paths and Macdonald functions We consider partition functions on the $N\times N$ square lattice with the local Boltzmann weights given by the $R$matrix of the $U_{t}(\widehat{sl}(n+1m))$ quantum algebra. We identify boundary states such that the square lattice can be viewed on a conic surface. The partition function $Z_N$ on this lattice computes the weighted sum over all possible closed coloured lattice paths with $n+m$ different colours: $n$ ``bosonic'' colours and $m$ ``fermionic'' colours. Each bosonic (fermionic) path of colour $i$ contributes a factor of $z_i$ ($w_i$) to the weight of the configuration. We show that $Z_N$ is a symmetric function in two alphabets $(z_1\dots z_n)$ and $(w_1\dots w_m)$. When $x_1\dots x_N$ are set to be equal to the box content of a skew Young diagram $\mu/\nu$ with $N$ boxes the partition function $Z_N$ reproduces the skew Macdonald function $P_{\mu/\nu}\left[wz\right]$.

ItemNo Preview AvailableThe mathematics of individualbased modelling: developing a realistic model of simple epithelial tissueGermano, Domenic Paul Joe ( 202305)Simple epithelia are the functioning components of many tissues found throughout the body. However, they are susceptible to disruption, which can lead to diseases such as cancer, asthma, cardiac disease, and viral infections. Before we can understand how these diseases occur, we must first understand how these tissues are normally maintained. Individualbased modelling is one such way to study simple epithelia. This thesis aims to gain a better understanding of the mathematics and mechanisms that underpin realistic individualbased models. We use these findings to develop a realistic model of simple epithelia. This research consists of two key parts. The first part focuses on understanding the fundamental mathematical constructions of individualbased models. In this research, we investigate three individualbased models of tissue dynamics: Overlapping Spheres, Voronoi Tessellations and Vertex Models. We investigate how particular modelling assumptions made at the tissue and cell boundaries affects both tissue growth and tissue collision. We find that all models are sensitive to their boundary description, with Overlapping Spheres models being highly sensitive to evolutionary timescale, tissue structures of Voronoi Tessellation models being highly sensitive to their tissue shape, and Vertex Models being the lest sensitive description. This research emphasises the importance of thorough mathematical understanding to undertake model selection for specific problems, as to ensure macroscopic tissue behaviours are not artefacts of model selection. Upon understanding the importance of model selection, we then consider the sensitivity of the Centrebased models of Overlapping Spheres and Voronoi Tessellation models. By investigating the models’ parameters, we demonstrate how they contain two independent time scales of tissue evolution. We also provide a guide for numerically solving the equations of motion and demonstrate how naive parameter choices can result in unstable behaviour. Finally, to ensure biologically realistic dynamics in the model, a degree of Brownian motion should be incorporated, unless a tissue maintains high cell renewal. After understanding the fundamental mathematics of individualbased models, the second part of this research introduces a novel threedimensional model of simple epithelia. Our description of the simple epithelia is deformable and consists of multiple layers. The movement of cells within the tissue is regulated by minimising a bending potential, cellcell adhesion, and cell viscosity. We demonstrate that this model is robust to tissue relaxation and dynamic homoeostasis while undergoing renewal. Lastly, we also show how the description is capable of maintaining the structure at dynamic homoeostasis under regeneration via cell migration and removal, and we show the model is comparable to that of a fixed geometry, without the need for the unrealistic limitations. Finally, we show how our novel model can describe tissues with curved surfaces, applying the model to describe spherical organoids under regimes of relaxation and renewal, showing that dynamic homoeostasis is maintained. We propose a novel extension that is capable of maintaining actively deforming structures, in specified regions within the tissue, to describe highly generalised tissue structures. We demonstrate that this extended model exhibits robustness under tissue relaxation and renewal, while undergoing active tissue deformations. Finally, we show our description of general simple epithelial can describe tissue regeneration via cell migration and removal, while undergoing active tissue deformations. The results and findings of this research will prove valuable to better understanding the mechanisms that contribute to simple epithelial tissue maintenance and homoeostasis within the human body.

ItemEconomic and Social Aspects of Heterogeneous Assembly LinesSato Michels, Adalberto ( 202310)Assembly lines with heterogeneous resources are widely present in manufacturing industries. The procedures to build a broad range of products employ various skilled workers or robots equipped with diverse tools. In order to remain competitive, it is crucial for a company to efficiently meet the market demand and reduce expenses at the same time. Since resources such as facilities, workforce wages, machines, and tools are quite costly, this requirement gives rise to the need to design economically viable lines. Conversely, we must take a myriad of technological and physical factors into account as well. For instance, assembly lines might appear in straight or Ushaped layouts, have continuous/(a)synchronous paces, operate with twosided or multimanned stations, manufacture multiple products in a mixedmodel fashion, and employ a specialised workforce with different capabilities. Furthermore, as demonstrated by the COVID19 pandemic scenario, possessing some resiliency for quick adaptations is a desirable feature. In addition, taking good care of employees fosters job satisfaction and reduces ergonomic risks, avoiding unnecessary turnover. Finally, it is also manageable to positively contribute towards societal issues, such as integrating workers with disabilities among ``conventional'' or robotic ones. Therefore, this thesis proposes three problems to investigate the complexities of economic and social aspects found in assembly lines with resource and workforce heterogeneity: the ResourceConstrained Assembly Line Balancing Problem (RCALBP), the Assembly Line Worker Assignment and Rebalancing Problem (ALWARP), and the Multimanned Assembly Line Worker Integration and Balancing Problem (MALWIBP) are herein studied and discussed. We develop MixedInteger Linear Programming (MILP) formulations for all three problems. They either aim at minimising the cycle time given limited resources or wages and facilities costs at the desired production rate. Moreover, a stateoftheart survey on Benders Decomposition (BD) approaches applied to the Assembly Line Balancing Problem (ALBP) is also presented. In the RCALBP and ALWARP, we explore resource and workforce heterogeneity appearing in realworld industrial applications. The former minimises cycle time given a limited number of stations and resources, whilst the latter aims at preserving jobs while minimising labour costs. We consider dedicated and alternative resource types for tasks, take scenarios with falling demands into account, and impose regularity metrics on workload reductions. These problems are mathematically modelled within a MILP framework, and benchmark instances are solved in commercial solvers along with case studies. As both RCALBP and ALWARP instances have very restrictive constraints, we can optimally solve large cases with commercial solvers by making problemspecific adjustments in the parameter tuning, as well as incorporating strong valid inequalities, variable reduction, and lower bounding techniques into the formulation. However, when the problem grows too complex, we may have to resort to heuristic approaches to obtain nearoptimal solutions in a reduced computational time. The MALWIBP examines the balancing of assembly lines with multimanned stations running on a heterogeneous workforce. This union creates a highly combinatorial problem: we must further link the already coupled decisions on assigning tasks to heterogeneous workers and workers to stations with task scheduling assessments. Thus, two heuristic solution procedures are developed, which tackle the problem with a hierarchical decomposition approach, showing that multioperated stations can reduce the assembly line's length even in the presence of a heterogeneous workforce. Lastly, inspired by the decomposition methods realm, a wellestablished exact strategy for solving large optimisation problems is surveyed. More specifically, a comprehensive literature review on applying classical and logicbased BD approaches to ALBP variations is inspected. As several literature contributions have recently employed BD algorithms to tackle ALBPs with practical extensions, this survey attempts to consolidate the current body of knowledge by providing a detailed literature review on each application's particular aspects and ideas. We summarise existing gaps to offer insights into the BD efficiency for combinatorial problems such as ALBPs from a managerial perspective and indicate a shift in the research trend. In concluding remarks, the contributions of the developed works are summarised and future research avenues are suggested for all problems.

ItemParisian Ruin with Random Delays for Spectrally Negative Lévy ProcessesNguyen, Duy Phat ( 202312)This thesis is devoted to studying Parisian ruin problems for spectrally negative Levy processes. The thesis consists of six chapters. Chapter 1 is a general introduction to spectrally negative Levy processes. We give definitions, examples and general properties of such processes. We also review the basics of stochastic calculus. Our exposition in this chapter often presents sketches of arguments rather than rigorous proofs. To compensate for the lack of rigour, we included references to more specialised texts where the proofs of the cited/used results can be found. Chapter 2 is devoted to the scale functions of spectrally negative Levy processes. We give basic definitions, examples and general properties. We also discuss some applications of scale functions. Chapter 3 presents previously known results of Parisian ruin theory and some related topics. We give the definitions and wellknown facts in these areas. Chapter 4 introduces a new interesting and natural extension to the Parisian ruin problem under the assumption that the risk reserve dynamics are given by a spectrally negative Levy process with trajectories of locally bounded variation. The novel feature of this extension is that the distributions of the lengths of the random implementation delays can depend on the deficit at the epochs when the risk reserve process turns negative, starting a new negative excursion. Moreover, this extension allows for the possibility of an immediate ruin when the deficit hits a certain subset. In this setting, we derive a closedform expression for the Parisian ruin probability and the joint Laplace transform of the Parisian ruin time and the deficit at that time of ruin. Chapter 5 is devoted to extending the results obtained in Chapter 4 to the case of spectrally negative Levy processes with trajectories of unbounded variation. Chapter 6 deals with the Parisian ruin time with arbitrary delays being independent of the deficit for the compound Poisson risk model. We show that the absolute distance between two Parisian ruin probabilities with different delays is bounded from above by the LevyProkhorov distance between the distributions of the two delays multiplied by a function of the initial reserve value. This means that the Parisian ruin probability with an arbitrary delay distribution can be approximated by the Parisian ruin probability with delay windows following a (finite) mixture of Erlang distributions, and the latter probability admits a closedform expression.

ItemMathematical approaches to pattern formation in dermatologyGilmore, Stephen. (University of Melbourne, 2005)

ItemAspects of mixed longitudinal growth analysisMatta, Alonso Alejandro. (University of Melbourne, 2010)This thesis presents practical approaches to the analysis of mixed longitudinal growth data. Longitudinal studies of the human population are specifically designed to investigate changes over a limited age range in a characteristic which is measured repeatedly for each study participant. This type of data poses several methodological challenges. First, models for the analysis of longitudinal data must recognize the relationship between the observations taken from each study participant. The mixed nature of the data calls for the use of random effects and variance and correlation structures for the within group errors. Secondly, the models must be flexible enough so that they can be easily differentiated for the timing of the population growth spurts. And thirdly, longitudinal growth data of human subjects is more often than not affected by the missing data problem. In practice, the missing data mechanism needs to be understood and taken into consideration when fitting the models. These aspects of mixed longitudinal growth analysis are covered in detail in this thesis using a comprehensive data set of repeated measures of human height of hundreds of Melbourne school children ranging form the ages of 5 to 18 years.

ItemPointwise axiomatic spectral theory in Banach algebrasLubansky, Raymond Alan. (University of Melbourne, 2008)

ItemThe detection and characterisation of complex spatiotemporal patterns of brain dynamics using fMRI, with an application to a study of motor learningDuff, Eugene Paul. (University of Melbourne, 2008)

ItemOn sample size determination for discrete dataGordon, Ian Robert. (University of Melbourne, 1993)